Proceedings of the II International Conference on Fundamental Interactions Pedra Azul, Brazil, June 2004 COSMOLOGY: TECHNIQUES AND OBSERVATIONS∗ Emilio Elizalde† Department of Mathematics, Massachusetts Institute of Technology 77 Massachusetts Ave, Cambridge, MA 02139-4307, USA. These lectures were addressed to nonspecialists willing to learn some basic facts, ap- proaches, tools and observational evidence which conform modern cosmology. The aim is also to try to complement the many excellent treatises that exists on the subject (an exhaustive treatment being in any case impossible for lack of time, in the lectures, and of space here), instead of trying to cover everything in a telegraphic way. We start by recalling in the introduction a couple of philosophical questions that have always upset inquiring minds. We then present some original mathematical approaches to investigate a number of basic questions, as the comparison of two point distributions (each point corresponding to a galaxy or galaxy cluster), the use of non-standard statistics in the analysis of possible non-Gaussianities, and the use of zeta regularization in the study of the contributions of vacuum energy effects at the cosmological scale. And we also summarize a number of important issues which are both undoubtedly beautiful (from the physical viewpoint) and useful in present-day observational cosmology. To finish, the reader should be warned that, for the reasons already given and lack of space, some fundamental issues, as inflation, quantum gravity and string theoretical fundamental ap- proaches to cosmology will not be dealt with here. A minimal treatment of any of them would consume more pages than the ones at disposal and, again, a number of excellent treatments of these subjects are available. arXiv:gr-qc/0409076v1 20 Sep 2004 Keywords: Cosmology; Vacuum Energy; Zeta Functions. 1. Introduction Cosmology is the study of the world we live in. It would be difficult to find a more noble endeavour for a human being —let aside perhaps from medicine— given our proven capacity to ‘understand’ things and events around us. To start, some already classic books, but very useful to everyone to get a basic knowledge of the cosmos are these in Ref.1 Here, let us just recall the several century long discussions, first on the issue of our Earth being flat or round, then about that same Earth being or not the center of the universe, and later the formulation of the universal law of gravitation by Newton, who extended thereby the observation of the behavior of an apple falling from a tree on the earth surface to build a universal law for the whole ∗Lectures given at the “II International Conference on Fundamental Interactions”, Pedra Azul, Brazil, June 6-12, 2004. †On leave from ICE, Consejo Superior de Investigaciones Cient´ıficas, and IEEC, Edifici Nexus, Gran Capit`a2-4, 08034 Barcelona, Spain. E-mail: [email protected], [email protected]. 1 2 E. Elizalde of the cosmos. This immediately rises to the thinking mind (at least) two questions. Why is the universe so ‘understandable’ to the human mind? • And how comes that mathematics are so useful in the formulation of the laws of • nature? Just recall the extreme simplicity of Newton’s formula, F = GMm/r2, and the usefulness of Leibniz’s calculus. Is this not very remarkable? Indeed, the question phrased by Eugene Wigner as that of the unreasonable effectiveness of mathematics in the natural sciences 2 is an old and intriguing one. It goes back to Pythagoras and his school, ca. 550 BC (“all things are numbers”), even probably to the sumerians, and maybe to more ancient cultures, which left no trace. Immanuel Kant said that “the eternal mystery of the world is its comprehensibility”a and Albert Einstein contributed also to this idea in his 1936 essay “Physics and Reality” where he elaborated on Kant’s statement. Also mathematical simplicity, and beauty, have remained for many years crucial ingredients when having to choose among different plausible possibilities. Those are for sure profound and far reaching ideas by some of the people who established landmarks in the long way towards our present understanding of the Cos- mos. However, one should be more humble and I would never dare to use words like “comprehend” or ‘understand’, but would rather replace them by describe, or mod- elize, in modern terms. Indeed, the fact e.g. that Newton’s formula is so extremely simple and far reaching does not mean at all that we understand the attraction of two massive bodies any better. Why do two bodies attract, and do not repel, each other? The only conclusion to be drawn is that Newton put us in possession of a very simple, useful, accurate, and universal model for ordinary gravity. In trying to answer questions as the last one and others, such as the nature of the mass of a body, and in trying to formulate an ultimately universal law for the whole Universe a lot of effort has been invested in the last 30 to 40 years. The results obtained have been extraordinary from the mathematical viewpoint (the new formalism itself and its mathematical applications, e.g. in algebraic geometry and knot theory). From the physical side the advance has been indeed consistent, but much less spectacular, and a unique theory of everything (TOE) is not in sight yet. In the last decade considerable effort has been put in observational cosmology with very rewarding results as, just to mention two of them, the construction of the first maps of the cosmos (first including thousands and now millions of galax- ies), and the discovery that the expansion of the universe we live in is accelerating. All indicates that this effort is going to continue, opening new perspectives for the awaited matching of the proposed mathematical models with the physical, observa- tional results, and also (not less important) for new jobs for cosmology students. aThe reader should pause and take some minutes to meditate such a profound and mysterious statement. The author cannot refrain from confessing that he had arrived to the same conclusion by himself alone, before reading Kant. When he first discovered Kant’s sentence, he was most deeply shocked. COSMOLOGY: TECHNIQUES AND OBSERVATIONS 3 These three lectures can only cover a very small part of what is known, or at least should be learnt, about cosmology. A personal bias is unavoidable and, far from trying to disguise it, the author’s purpose is to complement the many excellent trea- tises that exists on the subject, by touching upon some matters of his competence that are not so often discussed, and trying also to bring together things that appear disconnected in the existing literature. The first lecture will be on Mathematics as a tool to study the Universe. Among the new mathematics required to study physical processes since the appearance of Quantum Field Theory (QFT),3,4 there is the regularization or ‘summation’ of infinite series. In the cosmological setting and in brane and string theory or Quantum Gravity,5 the most elegant method to do this has proven to be zeta function regularization, introduced by S. Hawking in 1975 with exactly this purpose.6 A review of this technique7,8,9,10,11 will be presented in the first lecture, while a description of its uses in the calculation of the contri- bution of vacuum energy effects at the cosmological scale (e.g. in trying to explain the observed acceleration of the universe) will appear in the third. Also in the first lecture, an introduction of the formalism ordinarily used in Cosmology, as derived from the Einstein equations of General Relativity will be presented in some detail. The second lecture is devoted to some aspects of observational cosmology. We start by reviewing a number of important issues which are both extremely beautiful (from the physical viewpoint) and useful in present-day observational cosmology, such as the Hubble law, the Sunyaev-Zeldovich effect, the appearance of the Lyman- alpha forest structures, and the two variants of the Sachs-Wolfe effect. Then we present in some detail the mathematical techniques used for the purpose of the study of the large scale structure of our Universe. These include the analysis of point distributions (each point corresponding to a galaxy or galaxy cluster) and the use of non-standard statistics in the analysis of possible non-Gaussianities of the cosmic microwave background (CMB) temperature and matter fluctuations. Finally, as anticipated already, in the third lecture we will present some simple model in order to explain how quantum vacuum effects at cosmological scale can possibly contribute to the present value of the cosmological constant. For this purpose a further elaboration of the zeta regularization techniques, including some formulas originally derived by the author, will be necessary. There is no main conclusion to the paper, aside from the sum of the partial ones already suggested in the different sections. It is by now quite clear that the study of the Cosmos during the next decades will be most rewarding, not only intellectually or spiritually, but also in many more materialistic ways, for the many international collaborations already started or projected will undoubtedly require a large number of dedicated cosmologists in different tasks. Among them, to analyze and understand the observational results in the framework of existing and new ambitious theories, of which, for sure, there’ll still be plenty in the coming years. 4 E. Elizalde 2. Mathematics as a tool to study the Universe We now ellaborate on the issues rised in the Introduction. 2.1. Some disgression on divergent series The fact that the infinite series 1 1 1 s = + + + (1) 2 4 8 · · · has the value s = 1 is nowadays clear to any school child.